This paper presents the force coefficients (added mass and damping) for a circular cylinder oscillating horizontally in a uniformly stratified fluid of limited depth and in a continuously stratified fluid with a smooth pycnocline. The frequency-dependent added mass and damping are evaluated from Fourier transforms of the experimental records of impulse response functions. The stratification is shown to have a strong effect on the fluid–body interaction. It is found that, when the characteristic vertical extent of stratification (depth of uniformly stratified fluid or pycnocline thickness) decreases, the power radiated with internal waves is reduced and the maximum of the frequency spectrum of wave power shifts toward lower frequency. The results of experiments are compared with available theoretical predictions.

A fluctuating-force model is developed for representing the effect of the turbulent fluid velocity fluctuations on the particle phase in a turbulent gas–solid suspension in the limit of high Stokes number, where the particle relaxation time is large compared with the correlation time for the fluid velocity fluctuations. In the model, a fluctuating force is incorporated in the equation of motion for the particles, and the force distribution is assumed to be an anisotropic Gaussian white noise. It is shown that this is equivalent to incorporating a diffusion term in the Boltzmann equation for the particle velocity distribution functions. The variance of the force distribution, or equivalently the diffusion coefficient in the Boltzmann equation, is related to the time correlation functions for the fluid velocity fluctuations. The fluctuating-force model is applied to the specific case of a Couette flow of a turbulent particle–gas suspension, for which both the fluid and particle velocity distributions were evaluated using direct numerical simulations by Goswami & Kumaran (2010). It is found that the fluctuating-force simulation is able to quantitatively predict the concentration, mean velocity profiles and the mean square velocities, both at relatively low volume fractions, where the viscous relaxation time is small compared with the time between collisions, and at higher volume fractions, where the time between collisions is small compared with the viscous relaxation time. The simulations are also able to predict the velocity distributions in the centre of the Couette, even in cases in which the velocity distribution is very different from a Gaussian distribution.

The paper considers the refraction of a plane shock wave at an interface between two streams of different Mach number. Particular attention is paid to the irregular wave systems. It is found that when the interface is slow-fast, that is when the speed of sound a0 in the first or incident wave medium is less than the speed of sound aB in the second or transmitted wave medium, then there are two irregular systems, one being a double Mach reflexion type and the other being a four-wave confluence type. There are also two irregular systems when the refraction is fast-slow; these are a single Mach reflexion type and an expansion wave type. This last system has a central expansion wave when the flow is steady and a continuous band expansion wave when the flow is self-similar. Only two of the irregular wave systems have been observed experimentally in the fully developed state. Possible degeneracies are discussed.

The flow of a rotating, incompressible fluid over isolated topography whose non-dimensional height (i.e. topographic height divided by the mean fluid depth) is large compared with the Rossby number is studied. Attention is restricted to flow which is sufficiently shallow that the free-surface equations provide an adequate description. The flow is forced laterally by a specified upstream inflow (obtained from solutions of the zonally symmetric model equations) and by a prescribed surface stress. Dissipation is incorporated using a Rayleigh friction acting anti-parallel to the flow.

Steady-state solutions for uniform inflow on an f-plane are found for (a) linear viscous flow, (b) quasi-geostrophic flow with and without friction and (c) inviscid flow with and without a rigid lid. The presence of friction produces an upstream–downstream flow asymmetry over the obstacle and an associated topographic drag while inertial terms produce left-right (relative to an observer looking downstream) asymmetry. The blocking efficiency B (the percentage of the incident mass flux going around the obstacle rather than over it) of a Gaussian obstacle is largest (∼ 100%) for case (a) when viscous effects are small. In contrast quasi-geostrophic theory calculates no flow blocking (B ≡ O). For inviscid inertial theory, B ∼ 10% and is independent of the Rossby number. The presence of a free surface decreases the blocking for small-Rossby-number flow.

Numerical solutions of the appropriate initial, boundary-value problem for the complete model equations confirm these results and extend them to include the effects of (i) horizontal shear in the upstream inflow, (ii) the magnitude and shape of the topography, and (iii) variations in the Coriolis parameter (β-effect).

The direct-simulation Monte Carlo procedure for the modelling of rarefied gas flows on a digital computer has been used to generate representations of the simulated flows on a computer display unit. Photographs prepared from the displays illustrate the structure of the flows at the molecular level. The method has been applied to the steady hypersonic flow past circular cylinders, spheres, flat plates and blunt cones. These flows are represented either by several hundred typical molecular paths or by several thousand small arrows, each representing the position and velocity of a sample molecule at a particular instant. Colour was used to differentiate between undisturbed free-stream molecules, molecules that had struck the surface of the body and molecules that had been indirectly affected by the presence of the body. The method was also applied to the flow of a binary gas mixture past a circular cylinder, with colour used to differentiate between the components of the mixture.

Three-dimensional doubly diffusive convection in a closed vertically extended container driven by competing horizontal temperature and concentration gradients is studied by a combination of direct numerical simulation and linear stability analysis. No-slip boundary conditions are imposed on all six container walls. The buoyancy number $N$ is taken to be $-1$ to ensure the presence of a conduction state. The primary instability is subcritical and generates two families of spatially localized steady states known as convectons. The convectons bifurcate directly from the conduction state and are organized in a pair of primary branches that snake within a well-defined range of Rayleigh numbers as the convectons grow in length. Secondary instabilities generating twist result in secondary snaking branches of twisted convectons. These destabilize the primary convectons and are responsible for the absence of stable steady states, localized or otherwise, in the subcritical regime. Thus all initial conditions in this regime collapse to the conduction state. As a result, once the Rayleigh number for the primary instability of the conduction state is exceeded, the system exhibits an abrupt transition to large-amplitude relaxation oscillations resembling bursts with no hysteresis. These numerical results are confirmed here by determining the stability properties of both convecton types as well as the domain-filling states. The number of unstable modes of both primary and secondary convectons of different lengths follows a pattern that allows the prediction of their stability properties based on their length alone. The instability of the convectons also results in a dramatic change in the dynamics of the system outside the snaking region that arises when the twist instability operates on a time scale faster than the time scale on which new rolls are nucleated. The results obtained are expected to be applicable in various pattern-forming systems exhibiting localized structures, including convection and shear flows.

Normal modes of shallow water waves in a channel wherein the Coriolis parameter and the depth vary in the spanwise direction are investigated based on the conservation of the number of zeros in an eigenfunction. As a result, it is generally shown that the condition for transition modes (Kelvin modes and mixed Rossby-gravity modes) to exist, besides Rossby and Poincaré modes, is determined only by boundary conditions. A Kelvin mode is interpreted as a modification of a Kelvin wave or a boundary wave along a closed boundary, and a mixed Rossby-gravity mode as a modification of an inertial oscillation or a boundary wave along an open boundary. Transition modes appearing in edge and continental-shelf waves, equatorial waves and free oscillations over a sphere are systematically understood by applying the theory in this paper.

Vortex breakdown is a phenomenon inherent to many practical problems, such as leading-edge vortices on aircraft, atmospheric tornadoes, and flame-holders in combustion devices. The breakdown of these vortices is associated with the stagnation of the axial velocity on the vortex axis and the development of a near-axis recirculation zone. For large enough Reynolds number, the breakdown can be time-dependent. The unsteadiness can have serious consequences in some applications, such as tail-buffeting in aircraft flying at high angles of attack. There has been much interest in controlling the vortex breakdown phenomenon, but most efforts have focused on either shifting the threshold for the onset of steady breakdown or altering the spatial location of the recirculation zone. There has been much less attention paid to the problem of controlling unsteady vortex breakdown. Here we present results from a combined experimental and numerical investigation of vortex breakdown in an enclosed cylinder in which low-amplitude modulations of the rotating endwall that sets up the vortex are used as an open-loop control. As expected, for very low amplitudes of the modulation, variation of the modulation frequency reveals typical resonance tongues and frequency locking, so that the open-loop control allows us to drive the unsteady vortex breakdown to a prescribed periodicity within the resonance regions. For modulation amplitudes above a critical level that depends on the modulation frequency (but still very low), the result is a periodic state synchronous with the forcing frequency over an extensive range of forcing frequencies. Of particular interest is the spatial form of this forced periodic state: for modulation frequencies less than about twice the natural frequency of the unsteady breakdown, the oscillations of the near-axis recirculation zone are amplified, whereas for modulation frequencies larger than about twice the natural frequency the oscillations of the recirculation zone are quenched, and the near-axis flow is driven to the steady axisymmetric state. Movies are available with the online version of the paper.

Results are reported of an experimental investigation into the motion of a heavy cylinder free to move inside a water-filled drum rotating around its horizontal axis. The cylinder is observed to either co-rotate or, counter-intuitively, counter-rotate with respect to the rotating drum. The flow was measured with particle image velocimetry, and it was found that the inner cylinder significantly altered the bulk flow field from the solid-body rotation found for a fluid-filled drum. In the counter-rotation case, the generated lift force allowed the cylinder to freely rotate without contact with the drum wall. Drag and lift coefficients of the freely counter-rotating cylinder were measured over a wide range of Reynolds numbers, 2500 < Re < 25000, dimensionless rotation rates, 0.0 < α < 1.2, and gap to cylinder diameter ratios 0.003 < G/2a < 0.5. Drag coefficients were consistent with previous measurements on a cylinder in a uniform flow. However, for the lift coefficient, considerably larger values were observed in the present measurements. We found the enhancement of the lift force to be mainly caused by the vicinity of the wall.

Howarth's transformation is applied to the compressible laminar boundary-layer equations for the flow at a point of attachment on a general curved surface. It is shown that the boundary-layer equations yield similarity solutions for the case when the viscosity varies linearly with the temperature, the Prandtl number is unity and the surface is maintained at constant temperature. The resulting eighth-order boundary-value problem is solved numerically for various surface temperature conditions and for various values of b/a, where a and b are constants representing the local velocity gradients in the principal directions of flow at the edge of the boundary layer.

Flow and heat transfer properties of the similar solutions at both nodal and saddle-points of attachment are given in graphical and tabular form.

This paper deals with a specially designed slender ship moving steadily at a chosen supercritical speed in a suitable narrow shallow channel such that the wave resistance becomes zero. It is a sequel to Chen & Sharma (1997), which predicted theoretically the existence of a ship–channel configuration of zero wave resistance. It was derived from a two-soliton solution of the Kadomtsev–Petviashvili equation. The present theory comprises an improved shallow-water wave equation of Boussinesq type for the far field and an enhanced slender body approximation in the near field. Moreover, a physical model experiment performed in the Duisburg shallow water towing tank (VBD) is reported. It confirms that at the exact design condition the ship's reflected bow wave cancels the stern wave so completely that there are almost no waves behind the ship and the measured resistance is reduced so much that the wave resistance component is practically zero.

The transient pressure field produced behind a plane shock wave of arbitrary strength, which encounters a slender supersonic cone head on, is theoretically predicted. The analysis is restricted to small cone angles (in order that Mach-reflexion will occur at the surface) and to inviscid flow.

A conical-flow technique is utilized to transform the small-perturbation equations of time-dependent rotational motion of the fluid behind the deflected plane shock into time-independent equations. From these equations, a boundary-value problem for pressure alone, in the Mach-reflexion region, is derived which is solved numerically. Results for the small perturbation pressure field of the cone are contrasted with Smyrl's (1963) results, which hold for a moving wedge, for a range of values of cone and plane shock speeds.

A linear theory is developed for the spin-up of a compressible fluid, stratified by a spherical gravity field. Numerical results are obtained for the case of strong stratification (BruntVisl frequency N much greater than the rotation frequency 0). The interior flow is solved in terms of a set of angular eigenfunctions which have been obtained numerically. The principal result is that the spin-up is limited to a layer adjacent to the spherical boundary, the thickness of the layer being of the order of L(0/N), where L is the radius of the boundary. The solution is qualitatively similar to that found by Holton (1965), Walin (1969), and Sakurai (1969a, b) for a stratified fluid in a cylinder. The thickness of the spin-up layer diminishes with latitude , the variation being described roughly by the formula L0| sin |/N. For the case of slow continuous spin-up, the Ekman suction velocity has been calculated, and the results show that || = 24 is the dividing angle between suction (|| > 24) and blowing (|| < 24).

In part 1 (Woods 1979) generalized constitutive relations for the fluid stress p and the heat flux q were established by mean-free-path arguments. These relations are expected to hold over a wide range of Knudsen numbers K, but in the earlier paper, which presented the general theory, they were successfully tested only to O(K2), the order to which the Burnett constitutive equations are valid. It remains to verify the general theory at much higher K values, and to this end we have applied it to the propagation of forced sound waves in a rarefied monatomic gas.

Theories for such waves (based on the linearized Boltzmann equation) are available, and there are also experimental results for the speed and attenuation of waves up to K values of about 10. Our theory is in good agreement with experiment in the range 0 < K < 1. For K > 1 we obtain close agreement with experimental values of the wave speed, but as wave damping in this range is largely due to the unavoidable proximity of the wave transmitter – an effect not included in our calculations – we have found smaller values of the attenuation than obtaining in the experiments.

We use the adjoint method to compute sensitivity maps for the limit-cycle frequency and amplitude of the Bénard–von Kármán vortex street in the wake of a circular cylinder. The sensitivity analysis is performed in the frame of the semi-linear self-consistent model recently introduced by Mantič et al. (Phys. Rev. Lett., vol. 113, 2014, 084501), which allows us to describe accurately the effect of the control on the mean flow, but also on the finite-amplitude fluctuation that couples back nonlinearly onto the mean flow via the formation of Reynolds stress. The sensitivity is computed with respect to arbitrary steady and synchronous time-harmonic body forces. For a small amplitude of the control, the theoretical variations of the limit-cycle frequency predict well those of the controlled flow, as obtained from either self-consistent modelling or direct numerical simulation of the Navier–Stokes equations. This is not the case if the variations are computed in the simpler mean flow approach overlooking the coupling between the mean and fluctuating components of the flow perturbation induced by the control. The variations of the limit-cycle amplitude (that falls out the scope of the mean flow approach) are also correctly predicted, meaning that the approach can serve as a relevant and systematic guideline to control strongly unstable flows exhibiting non-small, finite amplitudes of oscillation. As an illustration, we apply the method to control by means of a small secondary control cylinder and discuss the obtained results in the light of the seminal experiments of Strykowski & Sreenivasan (J. Fluid Mech., vol. 218, 1990, pp. 71–107).

Transcritical flow of a stratified fluid past a broad localised topographic obstacle is studied analytically in the framework of the forced extended Korteweg–de Vries, or Gardner, equation. We consider both possible signs for the cubic nonlinear term in the Gardner equation corresponding to different fluid density stratification profiles. We identify the range of the input parameters: the oncoming flow speed (the Froude number) and the topographic amplitude, for which the obstacle supports a stationary localised hydraulic transition from the subcritical flow upstream to the supercritical flow downstream. Such a localised transcritical flow is resolved back into the equilibrium flow state away from the obstacle with the aid of unsteady coherent nonlinear wave structures propagating upstream and downstream. Along with the regular, cnoidal undular bores occurring in the analogous problem for the single-layer flow modelled by the forced Korteweg–de Vries equation, the transcritical internal wave flows support a diverse family of upstream and downstream wave structures, including kinks, rarefaction waves, classical undular bores, reversed and trigonometric undular bores, which we describe using the recent development of the nonlinear modulation theory for the (unforced) Gardner equation. The predictions of the developed analytic construction are confirmed by direct numerical simulations of the forced Gardner equation for a broad range of input parameters.

We present results from experiments designed to measure near-surface turbulence generated by rainfall. Laboratory experiments were performed using artificial rain falling at near-terminal velocity in a wind–wave channel filled with synthetic seawater. In this first series of experiments, no wind was generated and the receiving seawater was initially at rest. Rainfall rates from 40 to $190~\text{mm}~\text{h}^{-1}$ were investigated. Subsurface turbulent velocities of the order of $O(10^{-2})~\text{m}~\text{s}^{-1}$ are generated near the interface below the depth of the cavities generated by the rain drop impacts. The turbulence appears independent of rainfall rates. At depth larger than the size of the cavities, the turbulent velocity fluctuations decay as $z^{-3/2}$. Turbulent length scales also appear to scale with the size of the impact cavities. In these seawater experiments, a freshwater lens is established at the water surface due to the rain. At the highest rain rate studied, the resulting buoyancy flux appears to lead to a shallower subsurface mixed layer and a slight decrease of the turbulent kinetic energy dissipation. Finally, direct measurements and inertial estimates of the turbulent kinetic energy dissipation show that approximately 0.1–0.3 % of the kinetic energy flux from the rain is dissipated in the form of turbulence. This is consistent with existing freshwater measurements and suggests that high levels of dissipation occur at depths and scales smaller than those resolved here and/or that other phenomena dissipate a considerable amount of the total kinetic energy flux provided by rainfall.

The initial value problem for the motion of an intense, quasi-geostrophic, equivalent-barotropic, singular vortex near an infinitely long escarpment is studied in three parts. First, for times small compared to the topographic wave timescale the motion of the vortex is analysed by deriving an expression for the secondary circulation caused by the advection of fluid columns across the escarpment. The secondary circulation, in turn, advects the primary vortex and integral expressions are found for its velocity components. Analytical expressions in terms of integrals are found for the vortex drift velocity components. It is found that, initially, cyclones propagate away from the deep water region and anticyclones propagate away from the shallow water region. Asymptotic evaluation of the integrals shows that both cyclones and anticyclones eventually propagate parallel to the escarpment with shallow water on their right at a steady speed which decays exponentially with distance from the escarpment. Secondly, it is shown that for times comparable to, and larger than, the wave timescale, the vortex always resonates with the topographic wave field. The flux of energy in the topographic waves leads to a loss of energy in the vortex and global energy and momentum arguments are used to derive an equation for the distance (or, equivalently, the vortex velocity) of the vortex from the escarpment. It is shown that cyclones, provided they are initially within an O(1) distance (here a unit of distance is dimensionally equivalent to one Rossby radius of deformation) from the escarpment, drift further away from the deep water (i.e. toward higher ambient potential vorticity), possibly crossing the escarpment and accumulate at a distance of ≈1.2 on the shallow side of the escarpment. For distances larger than 1.2 there is essentially no drift of the vortex perpendicular to the escarpment. Anticyclones display similar behaviour except they drift in the opposite direction, i.e. away from the shallow water or toward lower ambient potential vorticity. Third, the method of contour dynamics is used to describe the evolution of the vortex and the interface representing the initial potential vorticity jump between the shallow and deep water regions. The contour dynamic results are in good quantitative agreement with the analytical results.

Some methods of formation of preforms for drawing of polarization-maintaining optical fibres are based on utilization of the surface tension of glass in the liquid state. Under the action of surface tension non-circular glass articles begin to flow, which results in formation of an anisotropic internal structure of the preforms. The hydrodynamic analysis of two such methods is given in the paper. Analytical solutions of the Stokes equations with linearized boundary conditions for the corresponding creeping surface-tension-driven flows of liquid glass are obtained. By means of these solutions a processing strategy may be predetermined with a view to a specific internal structure of the fibre, as well as to the required value of birefringence. The theoretical results are compared with experimental data and agreement is fairly good.